Methods and devices for controlling thermal conductivity and thermoelectric power of semiconductor nanowires

ABSTRACT

Methods and devices for controlling thermal conductivity and thermoelectric power of semiconductor nanowires are described. The thermal conductivity and the thermoelectric power are controlled substantially independently of the electrical conductivity of the nanowires by controlling dimensions and doping, respectively, of the nanowires. A thermoelectric device comprising p-doped and n-doped semiconductor nanowire thermocouples is also shown, together with a method to fabricate alternately p-doped and n-doped arrays of silicon nanowires.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of U.S. patent applicationSer. No. 12/175,027 filed on Jul. 17, 2008, which claims the prioritybenefit of U.S. Provisional Application No. 60/961,395 filed on Jul. 20,2007 for “High Performance Semiconductor Nanowire Thermoelectrics” byJames R Heath, Akram Boukai, Yuri Bunimovich, William A Goddard, andJamil Tahir-Kheli, the contents of all of which are herein incorporatedby reference in their entirety.

STATEMENT OF GOVERNMENT GRANT

The U.S. Government has certain rights in this invention pursuant toGrant No. DE-FG02-04ER46175 awarded by DOE, Grant No. CCF0524490 awardedby National Science Foundation and Grant No. N00014-07-1-0360 &N00014-06-1-0938 awarded by the Office of Naval Research.

BACKGROUND

Field

The present disclosure relates to thermoelectric devices and relatedfabrication methods. More in particular, it relates to methods anddevices for controlling thermal conductivity and thermoelectric power ofsemiconductor nanowires. It also relates to electric power generatorsand refrigerators based on semiconductor nanowires. More specifically,the semiconductor nanowires generate electric power wherever atemperature difference exists. They can also be used in reverse asrefrigerators whenever an electric current travels through thenanowires.

Related Art

Semiconductors are a class of materials whose electronic properties canbe tailored from metallic to insulating. This is accomplished through aprocess called “doping” whereby a small amount of impurity atoms areinjected into the semiconductor by ion implantation or diffusion.Semiconductors can either be made to conduct electrons or holes. Siliconis an example of a semiconductor. For example, silicon is made electronconducting by injection of phosphorus dopants whereas boron dopants makesilicon hole conducting. Dopants that render semiconductorselectron-rich are called n-type dopants and dopants that rendersemiconductors hole-rich are called p-type dopants. In general, theelectrical conductivity is proportional to the concentration of injecteddopants. Thus, the electronic properties can be precisely controlled bycontrolling the amount of injected dopants. The widespread use ofsemiconductors in the microelectronic industry is mainly due to theincredible control over their electronic properties.

Nanowires are a class of materials that have length scales for theirdiameter or width on the order of nanometers to tens of nanometers. Thenanometer scale is not easily accessible using conventional lithographicpatterning methods found in the microelectronic industry. Instead, thenanoscale may be made accessible using nanowire patterning or nanowirematerials growth methods. Typically, nanowires have an aspect ratio(length divided by width or diameter) that is equal to 10^(n) where ntypically varies from 1 to 5. The following literature providesrepresentative examples of the fabrication of semiconductor nanowiresand their doping:

-   1. Melosh, N. A. et al. Ultra-high density nanowire lattices and    circuits. Science 300, 112-115 (2003).-   2. Wang, D., Sheriff, B. A. & Heath, J. R. Complementary symmetry    silicon nanowire logic: Power-efficient inverters with gain. Small    2, 1153-1158 (2006).-   3. Morales, A. M. & Lieber, C. M. A laser ablation method for the    synthesis of semiconductor crystalline nanowires. Science 279,    208-211 (1998).    These three documents are incorporated herein by reference in their    entirety.

Thermoelectrics or thermoelectric materials are a class of materialsthat convert temperature differences into electricity and vice versa.Such materials utilize the Seebeck effect for power generation and thePeltier effect for refrigeration. In the Seebeck effect, a temperaturegradient across a thermoelectric material causes the diffusion ofcharged carriers across that gradient, thus creating a voltagedifference between the hot and cold ends of the material. Conversely,the Peltier effect explains the fact that when current flows through amaterial a temperature gradient arises because the charged carriersexchange thermal energy at the contacts. Therefore, thermoelectricmaterials can act as either electric power generators in the presence ofa temperature difference or as refrigerators when electric current issupplied.

Thermoelectrics are effectively engines that perform these functionswithout moving parts and they do not pollute. This makes them highlyreliable and more importantly attractive as clean power systems,especially at a time when global warming is a growing concern. Otherapproaches toward power generation or cooling such as fossil fuel basedengines emit pollution but are more efficient. As a result,thermoelectrics find only limited use because of their poor efficiency.

The efficiency of a thermoelectric material is determined by thedimensionless figure of merit,

${{ZT} = {\frac{S^{2}\sigma}{\kappa}T}},$where S is the thermoelectric power, defined as the thermoelectricvoltage, V, produced per degree temperature difference

${S = \frac{dV}{dT}},$σ is the electrical conductivity, κ is the thermal conductivity, and Tis the temperature. To maximize ZT, and thus the efficiency, S should belarge so that a small temperature difference can create a large voltage,a should be large in order to minimize joule heating losses, and κshould be small to reduce heat leakage and maintain the temperaturedifference. There is no intrinsic limit to how large ZT can be, but itis generally appreciated that a material with a ZT>1 constitutes athermoelectric of sufficient efficiency to have at least some practicalapplications. A thermoelectric with a ZT>3 would be transformative—forexample, thermoelectric-based cooling would replace existing compressioncycle refrigerators, and thermopower applications for heat recovery orenergy conversion would find widespread applications. Currently, thebest commercially available thermoelectric devices at room temperatureare alloys of Bi₂Te₃ and have a ZT of ˜1 which corresponds to a Carnotefficiency of ˜10%. Bi₂Te₃ is an exotic and expensive material tomanufacture and thus finding a thermoelectric material with a ZT>1 thatis earth abundant and cheap to process would allow more widespread useof thermoelectric devices. Finding a material with a ZT>1, however, ischallenging because optimizing one physical parameter often adverselyaffects another. The following literature provides reviews ofthermoelectric devices:

-   4. MacDonald, D. K. C. Thermoelectricity: An Introduction to the    Principles (Wiley, New York, 1962).-   5. Mahan, G., Sales, B. & Sharp, J. Thermoelectric materials: New    approaches to an old problem. Phys. Today 50, 42-47 (1997).-   6. Chen, G. et al. Recent developments in thermoelectric materials.    Int. Mater. Rev. 48, 45-66 (2003).-   7. Majumdar, A. Enhanced: Thermoelectricity in semiconductor    nanostructures. Science 303, 777-778 (2004).    These four documents are incorporated herein by reference in their    entirety.

In order to demonstrate an efficient thermoelectric, it is important tomeasure the three material parameters S, σ, and κ, and so calculate ZT.Such measurements can be carried out on nanowires using a variety ofon-chip thermometry and electrical leads. The nanowire electricalconductivity is measured by using a 4-point measurement to eliminatecontact resistance. For measurement of S and κ a temperature differenceis created across the ends of the nanowires by sourcing a DC currentthrough one of the resistive heaters. The resistance rise of eachthermometer is recorded simultaneously using a lock-in measurement asthe temperature is ramped upwards. The resistance of the thermometers istypically two orders of magnitude smaller than the nanowire array. Formeasurement of S, the thermoelectric voltage, as a response to thetemperature difference, is recorded using a nano-voltmeter. A differencemeasurement is used to determine κ The following literature providesrepresentative examples of measurements on thermoelectric devices:

-   8. Boukai, A., Xu., K. & Heath, J. R. Size-dependent transport and    thermoelectric properties of individual polycrystalline bismuth    nanowires. Advanced Materials 18, 864-869 (2006).-   9. Yu-Ming, L. et al. Semimetal-semiconductor transition in    Bi_(1-x)Sb_(x) alloy nanowires and their thermoelectric properties.    Applied Physics Letters 81, 2403-2405 (2002).-   10. Small, J. P., Perez, K. M. & Kim, P. Modulation of    thermoelectric power of individual carbon nanotubes. Physical Review    Letters 91, 256801 (2003).-   11. Li, S et al. Measuring thermal and thermoelectric properties of    one-dimensional nanostructures using a microfabricated device.    Journal of Heat Transfer 125, 881-888 (2003).-   12. Li, D. et al. Thermal conductivity of individual silicon    nanowires. Applied Physics Letters 83, 2934-2936 (2003)    These five documents are incorporated herein by reference in their    entirety.

In the following paragraphs, the challenges in optimizing the threethermoelectric materials parameters are delineated. In addition, therequirements for a practical thermoelectric device are described.Included in each description are the current state-of-the-art proceduresand systems for the best thermoelectric devices.

The thermoelectric power varies between different materials. In general,it has been found that the thermoelectric power is approximately 100times larger for semiconductors than metals. This is the main reasonthat semiconductors are the material of choice for thermoelectricdevices. The magnitude of the thermoelectric power for a semiconductordepends on the doping concentration. Typically, the thermoelectric poweris larger for low doped semiconductors and smaller for highly dopedsemiconductors. In addition, the thermoelectric power usually decreasesas the temperature is lowered for highly doped semiconductor metallicsystems. However, some semiconductors, such as silicon, have the uniqueproperty that their thermoelectric power increases when the temperatureis lowered. This behavior is due to phonon drag.

One physical phenomena that, in very specific systems, can increase S,is phonon drag. Phonon drag results when the phonons collide with eitherelectrons or holes and thus impart their momentum to the electroniccarriers. The phonons are in essence “pushing” the electrons and holesdown the temperature gradient. This results in an extra amount ofelectronic carriers diffusing down the temperature gradient and a largervoltage develops than would otherwise normally occur if phonon drag wasabsent. Phonon drag, therefore, leads to a larger thermoelectric power.Phonon drag has long been known to occur in low-doped semiconductorswhose electrical conductivity is poor. Therefore, phonon drag has notbeen successfully exploited in a practical thermoelectric devices sincethe low electrical conductivity reduces ZT. Increases in thethermoelectric power would be very beneficial as long as no degradationof the electrical conductivity occurs since the thermoelectric power issquared in the expression for ZT. The following literature providesrepresentative examples of observations of phonon drag on semiconductorthermoelectric devices:

-   13. Weber, L. & Gmelin, E. Transport properties of silicon. Applied    Physics A: Solids and Surfaces 53, 136-140 (1991).-   14. Herring, C. Theory of the thermoelectric power of    semiconductors. Physical Review 96, 1163-1187 (1954).-   15. Geballe, T. H. & Hull, G. W. Seebeck Effect in Silicon. Physical    Review 98, 940-947 (1955).-   16. Behnen, E. Quantitative examination of the thermoelectric power    of n-type silicon in the phonon drag regime. Journal of Applied    Physics 67, 287-292 (1990).-   17. Trzcinksi, R., Gmelin, E. & Queisser, H. J. Quenched Phonon Drag    in Silicon Microcontacts. Phys. Rev. Lett. 56, 1086-1089 (1986).    These five documents are incorporated herein by reference in their    entirety.

The electrical conductivity of a semiconductor can be controlled throughthe doping concentration of impurity atoms. A large doping concentrationwill result in a large electrical conductivity. In contrast, a lowdoping concentration will result in a low electrical conductivity. Also,a high doping concentration will result in a lower thermoelectric powerso that there is an optimal doping concentration that maximizes S²σ,otherwise known as the power factor. Most semiconducting thermoelectricdevices are doped to a concentration of 10¹⁹ cm⁻³. This is no easy taskfor commercially available thermoelectric devices, a majority of whichconsist of exotic materials. The doping concentration of silicon (andother relatively simple semiconductors such as germanium), on the otherhand, can easily be controlled with high precision. Silicon, therefore,is a promising candidate for highly efficient thermoelectrics since itspower factor can be optimized. Unfortunately, bulk silicon ischaracterized by a large thermal conductivity, and this limits the ZT ofsilicon to near 0.01. The small ZT precludes the use of bulk siliconthermoelectric devices from entering the commercial market.

The thermal conductivity varies widely for many thermoelectricmaterials. In general, good thermoelectrics have a thermal conductivitybelow 10 W m⁻¹ K⁻¹. Silicon, for example, has a thermal conductivity˜150 W m⁻¹ K⁻¹ at room temperature making it impractical for commercialuse. Commercial thermoelectrics based on Bi₂Te₃ materials have a thermalconductivity of 3 W m⁻¹ K⁻¹ or lower at room temperature. This value, incombination with its favorable power factor leads to a ZT of ˜1 at roomtemperature. Recently, several groups have used nanostructured materialsto increase ZT by using two-dimensional superlattices (i.e. layers ofthin films) and zero-dimensional “quantum dots” which have a reducedthermal conductivity relative to their bulk counterparts. However, thematerials used in these studies are expensive and rare, and it is notalways possible to achieve high efficiencies for both p- and n-typeconductors. It is not always straightforward to even prepare both p- andn-type conductors of these systems. Thermopower and thermocoolingapplications require both p- and n-type thermoelectric materials. Thefollowing literature provides representative examples of observations ofhigh ZT on semiconductor thermoelectric devices due to decreased thermalconductivity:

-   18. Venkatasubramanian, R. et al. Thin-film thermoelectric devices    with high room-temperature figures of merit. Nature 413, 597-602    (2001).-   19. Harman, T. C. et al. Quantum dot superlattice thermoelectric    materials and devices. Science 297, 2229-2232 (2002).-   20. Hsu, K. F. et al. Cubic AgPb_(m)SbTe_(2+m): Bulk thermoelectric    materials with high figure of merit. Science 303, 818-821 (2004).    These three documents are incorporated herein by reference in their    entirety.

Any practical thermoelectric device contains both p- and n-type dopedsemiconductor elements alternately connected electrically in series andthermally in parallel (as shown in FIG. 1A). One pair of p- and n-typedoped semiconductor elements connected in this manner is called athermocouple. To increase the output voltage, many thermocouples areconnected together. The voltage output is given by NV_(TE) where N isthe number of thermocouples and V_(TE) is the thermoelectric voltage ofone thermocouple. The following reference, incorporated herein byreference in its entirety, describes methods to fabricate p- and n-typethermoelectric elements that are connected electrically in series andthermally in parallel:

-   21. Snyder, G. J. et al. Thermoelectric microdevice fabricated by a    MEMS-like electrochemical process. Nature Materials 2, 528-531    (2003)

It is often difficult to dope a semiconductor both p- and n-type. It canalso be difficult to precisely control the doping concentration.Silicon, germanium, and their alloys, however, have a distinct advantageover other semiconductors because they can easily be doped p- andn-type. Moreover, repeated and controlled doping of silicon nanowireshas been demonstrated. The following reference, incorporated herein byreference in its entirety, describes methods to dope siliconsemiconductor nanowires both p- and n-type:

-   22. Wang, D., Sheriff, B. A. & Heath, J. R. Complementary symmetry    silicon nanowire logic: Power-efficient inverters with gain. Small    2, 1153-1158 (2006).

In summary, the majority of bulk semiconductors are typically poorthermoelectrics either due to their large thermal conductivity and/orsmall electrical conductivity. Also, current thermoelectric devices donot take advantage of phonon drag effects. In a typical thermoelectric,the three material parameters thermal conductivity, electricalconductivity, and thermopower are interdependent.

SUMMARY

According to a first aspect, a method of controlling thermalconductivity and thermoelectric power of a material while substantiallymaintaining electrical conductivity of said material is provided,comprising: providing semiconductor nanowires as said material;controlling the thermal conductivity of the semiconductor nanowiressubstantially independently of the electrical conductivity of thesemiconductor nanowires by controlling dimensions of the semiconductornanowires; and controlling the thermoelectric power of the semiconductornanowires substantially independently of the electrical conductivity ofthe semiconductor nanowires by controlling doping of the semiconductornanowires.

According to a second aspect, a thermoelectric device comprising p-dopedand n-doped semiconductor nanowire thermocouples is provided, saidp-doped and n-doped semiconductor nanowire thermocouples being connectedelectrically in series and thermally in parallel.

According to a third aspect, a method to fabricate alternately p-dopedand n-doped arrays of silicon nanowires is provided, comprising:providing a silicon-on-insulator (SOI) substrate; insulating a topsilicon layer of the SOI substrate; coating the insulated top siliconlayer with photoresist; patterning the photoresist into a series ofisolated, alternately p-doped and n-doped, regions separated byinsulator; removing the insulator between the isolated regions; etchingseparation regions under the removed insulator up to an insulator layerof the SOI substrate; removing the photoresist; patterning a pluralityof circles on top of each isolated region; etching the isolated regions,the plurality of circles for each isolated region acting as a mask, toobtain, for each isolated region, a nanowire array comprising pluralityof nanowires having same diameter as the circles, thus formingalternately p-doped and n-doped arrays of silicon nanowires.

Further embodiments are present in the specification, drawings andclaims of the present application.

Applicants have combined experiment and theory to demonstrate thatsemiconductor nanowires can be designed to achieve enhancements inthermoelectric efficiency, and have shown that the temperature ofmaximum efficiency may be tuned by changing the doping and the nanowiresize. Theory indicates that similar improvements should be achievablefor other semiconductor nanowire systems because of phonon effects.These nanowire thermoelectrics may find applications related to on-chipheat recovery, cooling, and power generation. Additional improvementsthrough further optimization of nanowire size, doping, composition,etc., should be possible.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A and 1B illustrate a thermoelectric device based on bulk p- andn-type thermocouples alternately connected electrically in series andthermally in parallel.

FIG. 2 shows a schematic of an embodiment of the present disclosure,where p- and n-type thermocouples alternately connected electrically inseries and thermally in parallel as in FIG. 1A are shown. However, eachthermocouple element comprises semiconductor nanowires rather than abulk piece of material.

FIGS. 3A-3C show scanning electron micrographs of a device architectureused to quantify ZT of the silicon nanowires in accordance with thedisclosure.

FIG. 4, panels a and b, show the thermal conductivity ratio of bulksilicon to nanowires and thermoelectric power squared data for 20 nmwide and 10 nm wide silicon nanowires. The thermal conductivity andthermoelectric power are used to calculate ZT.

FIGS. 5A and 5B show thermal conductivity of the 20 nm and 10 nm siliconnanowires and thermoelectric power of 20 nm nanowires for various dopingconcentrations.

FIG. 6 shows agreement of the theoretical fit to experimental data ofthe thermoelectric power. The theoretical fit takes into accountcontributions from phonon drag.

FIG. 7 shows electrical conductivity data for 20 nm and 10 nm widenanowires along with a bulk silicon reference sample. The electricalconductivity is used to calculate ZT.

FIG. 8 shows the ZT data for a representative set of 20 nm and 10 nmwide silicon nanowires.

FIGS. 9A-9F show a micrograph of vertically standing silicon nanowiresand depict a method to create a functioning thermoelectric circuit withp- and n-type nanowires. The example used here is based on siliconnanowires.

DETAILED DESCRIPTION

For the very simple system of silicon nanowires, Applicants found thatby tuning the nanowires' dimensions, the thermal conductivity isdramatically reduced, while maintaining a high electrical conductivity.Furthermore, Applicants have also found that the thermopower in highlydoped (good electrically conducting) nanowires can be increased furtherby phonon drag. The thermopower depends upon nanowire diameter, nanowirelength, and impurity doping. The thermal conductivity depends upon justnanowire diameter. Thus, the present application provides athermoelectric device that removes the interdependency of the threethermoelectric material parameters that govern ZT. Applicants usedsilicon nanowires as an example to engineer an excellent thermoelectricmaterial. Silicon is just one example of a semiconducting nanowirematerial that can be used to fabricate a high efficiency thermoelectricdevice. Other semiconductor materials such as germanium and alloys ofsilicon and germanium are likely to increase ZT when fabricated intonanowire form. In general, Applicants have found that any semiconductornanowire system for which the control of nanowire dimension and impuritydoping level is possible will show an increase in efficiency.

In other words, Applicants have found new physics that emerges at thenanoscale, and have harnessed such physics to transform a material thatis a poor thermoelectric into an efficient thermoelectric as describedbelow.

According to some of the embodiments of the present disclosure, methodsfor fabricating silicon nanowire thermoelectric devices with ZTvalues >1 are described below. As a demonstration, single-crystallinesilicon nanowires were fabricated using the Superlattice NanowirePattern transfer (SNAP) process (see Reference 1 mentioned above). Thenanowires were doped p-type using a boron containing spin-on dopant.Electron-beam lithography (EBL) was used to create Ti/Pt electrodes forthe electrical contacts.

According to one of the embodiments of the present disclosure, efficientthermoelectric performance from the single component system of siliconnanowires for cross-sectional areas of 10 nm×20 nm and 20 nm×20 nm hasbeen achieved. Silicon is a viable commercial material due to itsoverabundance relative to other semiconductor materials in the earth'scrust. In addition, the chemistry of silicon is well understood. Byvarying the nanowire size and impurity doping levels, ZT valuesrepresenting an approximately 100-fold improvement over bulk silicon areachieved over a broad temperature range, including a ZT˜1 at 200K.Independent measurements of S, a, and κ, combined with theory, indicatethat the improved efficiency originates from phonon effects. Theseresults are expected to apply to other classes of semiconductornanomaterials.

Applicants' observed high ZT for silicon nanowires occurs because κ issharply reduced below the minimum thermal conductivity, κ_(min)=1 W m⁻¹K⁻¹ at room temperature, of bulk amorphous silicon. Also, the phonondrag component of the thermoelectric power, S_(ph), becomes large.Below, Applicants show that S_(ph) increases due to a 3D to 1Ddimensional crossover of the phonons participating in phonon drag, andto decreasing κ.

The majority of bulk semiconductors are typically poor thermoelectricseither due to their large thermal conductivity and/or small electricalconductivity. Also, current thermoelectric devices do not take advantageof phonon drag effects. As described below, embodiments of the presentdisclosure provide for the fabrication of highly efficientthermoelectric devices based on semiconductor nanowires. As a specificexample, results of the three thermoelectric material parameters forp-type silicon nanowires are discussed. The unique feature of thesilicon nanowires is that they have a dramatically reduced thermalconductivity and exploit the effect of phonon drag. Embodiments of thepresent disclosure, therefore, provide for the fabrication ofthermoelectric nanowire devices with ZT values ≥1. Embodiments of thepresent disclosure may also allow for thermoelectric nanowire devices tobe made from a broad class of semiconductor materials. These aresemiconductors that allow for control of their doping level so that thethree thermoelectric material parameters can be precisely controlled.Embodiments of the present disclosure may also provide for thefabrication of vertically standing p- and n-type semiconductor nanowiresconnected electrically in series and thermally in parallel.

FIG. 1A shows how a typical thermoelectric device is constructed usingp- and n-type thermoelectric elements. The elements are alternatelyconnected electrically in series and thermally in parallel. The p- andn-type elements should be connected in this manner to ensure that theoutput voltage is maximized. Therefore, thermoelectric materials shouldbe optimized for both p- and n-type doping. As the top electricallyinsulating yet thermally conductive plate is made hotter than the bottomplate, electrons and holes diffuse across the temperature gradient andan electric voltage is produced. This voltage can be used to supplyelectric power. FIG. 1B shows the equivalent circuit of thermoelectricdevice in FIG. 1A where N is the number of p-n thermocouples. Each p-and n-type element is modeled as a voltage source that supplies avoltage=SΔT. A source resistance R_(p) and R_(n) is associated with eachp-type element and n-type element respectively. The total output voltageof this thermoelectric module is =NΔT(S_(n)+S_(n)).

FIG. 2 is a schematic of one of the embodiments of the presentdisclosure. Semiconductor nanowires (shown as thin lines) are connectedelectrically in series and thermally in parallel and are alternatelydoped p- and n-type to form a series of thermocouples. This structuretakes advantage of the dramatically reduced thermal conductivity andincreased thermopower due to phonon drag to increase ZT>1.

FIGS. 3A-3C show scanning electron micrographs of the device utilized tomeasure the thermoelectric power and electrical and thermal conductivityof silicon nanowire arrays.

FIG. 3A is a device structure used for measurement of the threethermoelectric material parameters: thermal conductivity, electricalconductivity, and thermopower. The electron micrograph shows an image ofa suspended platform with silicon nanowires and all electricalconnections. The central area (10) is the silicon nanowire array, whichis not resolved at this resolution. According to one of the embodimentsof the present disclosure, the silicon nanowires are made using the SNAPprocess. The 4-lead electrodes (20, 30, 40, 50) are utilized forthermometry to measure the temperature difference across the nanowirearray to obtain values for the thermopower. That thermal gradient isestablished with either of the two Joule heaters (60, 70). Theelectrodes (20, 30, 40, 50) and (80, 90) are combined to carry out4-point electrical conductivity measurements on the nanowires. Theregion (100) underlying the nanowires and the electrodes 150 nm thickSiO₂ insulator that is sandwiched between the top Si(100) single crystalfilm from which the nanowires are fabricated, and the underlying siliconwafer. The underlying silicon wafer etched back to suspend themeasurement platform, placing the background of this image out of focus.

FIG. 3B shows a low resolution micrograph of the suspended platform. Theelectrical connections radiate outwards and support the device.

FIG. 3C shows a high resolution image of an array (110), e.g., of 20 nmwide silicon nanowires with a metal electrode (120), e.g., a Ptelectrode.

FIG. 4, panels a and b, show factors contributing favorably to ZT forvarious silicon nanowires.

FIG. 4, panel a, shows the temperature dependence of the thermalconductivity (κ), presented as K_(Bulk)/K_(NW) to highlight theimprovement that the reduction of κ in nanowires due to reductions insize lends to ZT. K_(Bulk) values, which are slightly below the truebulk value for silicon, are taken from an identically measured 520 nm×35nm sized film. The thermal conductivity of the 10 nm wide siliconnanowires is smaller than the minimum thermal conductivity of bulkamorphous silicon. The inset in FIG. 4, panel a, shows micrographs showthe region of the device containing the nanowires before (top) and afterthe XeF₂ etch to remove the nanowires.

FIG. 4, panel b, shows the temperature dependence of S² for 20 nm widesilicon nanowires at various p-type doping concentrations (indicated onthe graph). Note that the most highly doped nanowires (lowest line)yield a thermoelectric power similar to that of bulk silicon doped at alower level. For nanowires doped at slightly higher and slightly lowerconcentrations than the bulk, S is peaked near 200K. This is aconsequence of the one-dimensional nature of the silicon nanowires.

For all but the most highly doped nanowires, S peaks near 200K. Thispeak is unexpected: similarly doped bulk silicon exhibits a gradualdecrease in S as T is reduced (second trace from the bottom). For T<100Ka peaked S(T) is observed for metals and lightly doped semiconductorsand is due to phonon drag.

Embodiments of the present disclosure exploit phonon drag effects.Phonon drag is generally assumed to vanish with decreasing sampledimensions because the phonon path-length is limited by the sample size.This appears to eliminate phonon drag as the reason for the peak in ournanowires. Below is shown that the phonon wavelengths participating indrag are on the order of or larger than the wire width. This leads to a3D to 1D dimensional crossover of these modes and removes thecross-sectional wire dimensions from limiting the phonon mean path (seeFIG. 5 discussed below). The nanowire boundaries are incorporated intothe 1D mode and are not an obstacle to phonon propagation. Therefore,the limiting size becomes the wire length (˜1 μm) and phonon drag“reappears” at very small dimensions.

In addition, classical elasticity theory is valid for the phononwavelengths considered here, leading to thermoelastic damping of soundwaves proportional to κ. Thus S_(ph) is further enhanced due to theobserved reduced thermal conductivity κ. It might appear that elasticitytheory leads to a contradiction because κ is proportional to the meanphonon lifetime. If the phonon lifetimes increase as stated above, thenκ should also increase. The resolution is that the elasticity expressionis only valid for long wavelength modes.

Below, the electronic and phonon contributions, S=S_(e)+S_(ph) to thethermoelectric power are considered separately for the nanowire data atT>200K. Charge carriers dissipate heat to the lattice through a processthat first involves momentum conserving (non-dissipative)electron-phonon collisions. The phonons that contribute to phonon dragcannot have a wavelength shorter than λ_(min), which is determined bythe size of the Fermi surface. Phonon drag is observed in metals only atlow T because the Fermi surface is large and the heat carrying shortwavelength phonons have short lifetimes. At low T (<20K), S_(ph)˜T³ fromthe phonon specific heat (˜T³). For kT>>Θ_(Debye), the specific heatbecomes constant and the number of phonons available for phonon-phononscattering is ˜T leading to S_(ph)˜1/T.

For p-type silicon, the holes are near the valence band maximum. Thephonon drag modes are acoustic with largest wavevector,k_(ph)=2k_(fermi)=0.2 Å⁻¹ (for impurity doping 3×10¹⁹ cm⁻³). Theshortest wavelength is λ_(ph)=2π/k=31 Å. Umklapp (non-momentumconserving) phonon-phonon scattering processes determine the rate ofphonon heat dissipation. The Debye energy (Θ_(D)) sets the energy scalefor Umklapp scattering. The number of Umklapp phonons available todissipate the long wavelength phonons is given by the Bose-Einsteinfunction

$N_{U} = \frac{1}{e^{\Theta_{D}/T} - 1}$leading to a scattering rate 1/Σ_(ph)˜N_(U). When T>>Θ_(D), 1/Σ_(ph)˜T.Since Θ_(Debye)=640K for silicon, the full Bose-Einstein expressionshould be applied for T≤350K.

The electronic contribution (S_(e)) is estimated from the Mott formula

${{S_{e}(T)} = {{\frac{\pi^{2}k^{2}T}{3e}\left( \frac{{\partial\ln}\mspace{11mu}{\sigma(ɛ)}}{\partial ɛ} \right)} \approx {\left( {283\mspace{14mu}\mu\;{V/K}} \right)\left( {{kT}/E_{f}} \right)}}},$where the conductivity derivative equals the reciprocal of the energyscale over which it varies (the Fermi energy E_(f)). Assuming holedoping occurs in the heavier silicon valence band (mass 0.49), thisleads to E_(f)=0.072 eV=833K and k_(f)=0.1 Å⁻¹ for n=3×10¹⁹ cm⁻³. ThusS_(e)(T)=aT where a=0.34 μV/K².

The T>200K thermoelectric power data of the 20 nm wire (doping n=3×10¹⁹cm⁻³) fitsS=S _(e) +S _(ph) =aT+b[exp(Θ_(D) /T)−1]where a, b and Θ_(D) are varied to obtain the best fit (see FIG. 6). Thecoefficients are a=0.337 μV/K², b=22.1 μV/K, and Θ_(Debye)=534 K. Thecoefficient a is almost identical to our estimate of 0.34 μV/K². Thusphonon drag explains the observed thermoelectric power. Consistent withmeasurements of phonon drag in bulk silicon, S in our nanowiresincreases significantly at lighter doping.

The phonon drag contribution to S is of the form

${\left. S_{ph} \right.\sim\left( \frac{\tau_{ph}}{\mu\; T} \right)}.$Σ_(ph), the phonon lifetime, is ˜1/κ from elasticity theory. μ is theelectron mobility. ZT scales as (neglecting S_(e))

${\left. S_{ph} \right.\sim\frac{1}{\mu\; T\;\kappa}},\mspace{14mu}{{\left. \sigma \right.\sim n}\;\mu},{\left. {ZT} \right.\sim\frac{n}{\mu\; T\;\kappa^{3}}},$leading to increased ZT with decreasing mobility. This is opposite theconclusion reached when considering only S_(e).

Embodiments of this disclosure may allow dramatically reduced thermalconductivities in semiconductor nanowires over bulk values.

FIG. 5A shows the dramatically reduced thermal conductivity of siliconthermoelectric nanowires which increases ZT. The thermal conductivity isreduced due to phonon effects that are present only at the nanoscaledimensions used here. The 10 nm wide silicon nanowires have a thermalconductivity below that of the minimum thermal conductivity of bulkamorphous silicon, κ_(min). Conventional wisdom states that thermalconductivity of silicon cannot be below κ_(min). However, the derivationof κ_(min) is based on several assumptions. First, the derivation ofκ_(min) assumes that the minimum path length of wavelength A phonons is(½)λ and that the phonons are described by the Debye model using bulksound speeds with no optical modes. The (½)λ value is an order ofmagnitude estimate and is difficult to determine precisely in analogy tothe minimum electron mean free path used to calculate theMott-Ioffe-Regel σ_(min). Also, κ_(min) is proportional to thetransverse and longitudinal acoustic speeds of sound. These are reducedin our nanowires at long wavelengths because the modes become 1D. Theratio of the 1D to 3D longitudinal speeds of sound is[(1+v)(1−2v)/(1−v)]^(1/2)=0.87 where v=0.29 is the Poisson ratio ofsilicon. The transverse acoustic speed goes to zero at long wavelengthsince co ω˜k²d where d is the nanowire width. Thus the bulk κ_(min)estimate above is invalid for our nanowires and values smaller thanκ_(min) are attainable.

FIG. 5B shows further evidence for phonon drag. The phonon drag effectincreases as semiconductor doping concentrations decreases and thusincreases ZT. Phonon drag effects are typically only seen in bulkmaterials. Phonon drag effects “re-appear” at the nanoscale dimensionstudied here. The doping level of the low-doped sample (upper curve) wasestimated from nanowire conductivity measurements, instead of 4-pointconductivity measurements of the silicon-on-insulator film from whichthe nanowires were fabricated. As the doping is decreased, the measuredS increases dramatically. This is consistent with phonon drag observedin lightly doped semiconductors. At lower doping, the Fermi surface issmaller and long wavelength acoustic phonons absorb the electronmomentum. Since long wavelength phonons have longer lifetimes, thephonon drag contribution to the thermopower increases as the dopingdecreases.

FIG. 6 shows the close agreement of the theoretical calculation to theexperimental data. The thermoelectric power calculation is plotted alongwith experimental data (black points) from a 20 nm wide silicon nanowirep-type doped at 3×10¹⁹ cm⁻³. The top curve is the fitted expression forthe total thermoelectric power S_(e)+S_(ph). The middle curve is thephonon contribution S_(ph) and the bottom line is the electronic termS_(e) arising from the fit. The fit has maximum error 6.1 μV/K and rmserror 1.8 μV/K. The experimental error bars at 150, 200, and 225K aresmaller than the data points. The data points above the bottom line areexperimental values for bulk wires (doping 2×10²⁰ cm⁻³) (crosses), 10 nmnanowires (doping 7×10¹⁹ cm⁻³) (diamonds), and 20 nm wires (doping1.3×10²⁰ cm⁻³) (triangles) where only a linear T electronic contributionwas found. This data is close to the extracted electronic contributionfrom the black data points (blue line) and shows that the fitted linearterm is reasonable. The drop in S to 0 as T→0 occurs because the phononmean free path reaches the sample size and the specific heat→0 due tothe third law of thermodynamics. The inset shows the character of a 3Dbulk longitudinal acoustic phonon mode (top) and a 1D mode when thewavelength is larger or on the order of the width. The 1D modeincorporates the existence of the boundary by transverse expansion(compression) for longitudinal compression (expansion). The ratio of thetransverse strain to the longitudinal strain is the Poisson ratio (0.29for silicon).

FIG. 7 shows representative electrical conductivity data for thesingle-crystal silicon nanowires and microwires; p-type doping levelsare indicated. All nanowires are 20 nm in height. The electrical currentcarrying capacity of the nanowires is large. Therefore, the electricalconductivity is nearly unaffected by the size reduction from bulk to thenanoscale. This fact helps to maintain the large ZT values.

FIG. 8 shows that silicon thermoelectric nanowires have largeefficiencies. In fact their efficiency is nearly 100 times larger thanbulk. The figure shows temperature dependence of the thermoelectricefficiency, ZT, for two different groups of nanowires. The crosssectional area of the nanowires, and the p-type doping level, are givenon the graph. The 20 nm wide nanowires have a thermoelectric power thatis dominated by phonon contributions, and a ZT value ˜1 is achieved near200K. The smaller (10 nm wide) nanowires have a thermoelectric powerthat is dominated by electronic contributions. The ZT at 350K iscalculated using the thermal conductivity value for the 10 nm nanowiresat 300K. The error bars represent 95% confidence limits.

FIG. 9A shows how a micrograph of a vertical array of siliconsemiconductor nanowires. The nanowires have a diameter of 20 nm and arefabricated by the SNAP process. The aspect ratio of the nanowires iscurrently 4, but this can be increased.

FIGS. 9B-9E show how a vertical array of nanowires can be doped into p-and n-type elements that are alternately connected electrically inseries and thermally in parallel. The example used here is based on asilicon nanowire thermoelectric device.

FIG. 9B shows a cross-section of the starting material for thethermoelectric device which is an un-doped silicon-on-insulatorsubstrate. The electrical insulator is SiO₂. The top layer of silicon isseveral microns thick and will eventually be transformed into an arrayof alternately doped p- and n-type nanowires that are connectedelectrically in series and thermally in parallel. The very most topsilicon layer is oxidized to a thickness of 100 nm and then coated witha photoresist. The oxide will act as an etch mask in a subsequent deepreactive ion etching (DRIE) step. Photolithography is then used topattern the photoresist into a series of isolated square featuresseveral microns long on each side. Then the 100 nm of oxide in betweenthe photoresist squares is removed using a buffered oxide etch of 1 parthydrofluoric acid and 5 parts water. DRIE is then performed to etch allthe way down to the buried SiO₂ insulator layer. This step electricallyisolates the silicon squares underneath the photoresist. The photoresistis then removed in acetone.

FIG. 9C shows a top view of each square alternately doped p- and n-typeusing photolithography masking and ion implantation techniques.

FIG. 9D shows a top view of round metallic circles 50 nm or smaller indiameter patterned on top of each square through SNAP and nano-imprintlithography techniques. The metallic circles act as etch masks for thesubsequent etching of the top silicon layer using DRIE or wet etching toform the nanowires.

FIG. 9E shows a top view of how the nanowires are formed. The nanowirescan be fabricated using a DRIE etch or wet etching. The etching stopswhen the exposed silicon (the silicon not directly underneath themetallic circles) is completely removed. The nanowires will have thesame diameter as the metallic circles. The device now comprisesalternately doped arrays of p- and n-type silicon nanowires.

FIG. 9F shows a top view of the metallized top contacts (bottom contactsnot shown). Metal contacts such as titanium and platinum electricallyconnect the alternately doped arrays of p- and n-type silicon nanowiresin series and also allows them to be connected thermally in parallel.

Accordingly, what has been shown are semiconductor nanowirethermoelectric devices and related fabrication methods. While thedevices and methods have been described by means of specific embodimentsand applications thereof, it is understood that numerous modificationsand variations could be made thereto by those skilled in the art withoutdeparting from the spirit and scope of the disclosure. It is thereforeto be understood that within the scope of the claims, the disclosure maybe practiced otherwise than as specifically described herein.

The invention claimed is:
 1. A thermoelectric device, comprising: ananowire array comprising an individual nanowire consisting essentiallyof doped silicon, wherein the individual nanowire comprises asemiconductor, and wherein the individual nanowire has a diameter up toabout 20 nanometers (nm) and a length that is at least about 4 times thediameter of the individual nanowire, wherein the individual nanowire isdoped p-type or n-type, but not both, and has a doping concentrationfrom about 3×10¹⁹ cm⁻³ to 2×10²⁰ cm⁻³, and wherein the individualnanowire has a thermoelectric efficiency that is at least one hundredtimes greater than a thermoelectric efficiency of bulk silicon, whereinthe thermoelectric efficiency of the individual nanowire is greater thanor equal to about 0.31 over a temperature range from about 100 K to 200K.
 2. The thermoelectric device of claim 1, wherein the individualnanowire is about 20 nm wide and p-type doped at a concentration ofabout 7×10¹⁹ cm⁻³.
 3. The thermoelectric device of claim 2, wherein anelectric conductivity of the individual nanowire changes less than 1,000Ω⁻¹ cm⁻¹ across a temperature range of 50K to 200K.
 4. Thethermoelectric device of claim 1, wherein the individual nanowire isabout 20 nm wide and p-type doped at a concentration of about 1.3×10²⁰cm⁻³.
 5. The thermoelectric device of claim 4, wherein an electricconductivity of the individual nanowire changes less than 1,000 Ω⁻¹ cm⁻¹across a temperature range of 50K to 200K.
 6. The thermoelectric deviceof claim 1, wherein the thermoelectric efficiency of the individualnanowire is maximal at a temperature of about 200 K.
 7. Thethermoelectric device of claim 1, wherein the diameter is from about 10nm to 20 nm.
 8. The thermoelectric device of claim 1, further comprisingelectric contacts in electrical communication with the nanowire array.9. The thermoelectric device of claim 8, wherein the electrical contactsinclude platinum.
 10. The thermoelectric device of claim 8, wherein theelectrical contacts include titanium.
 11. The thermoelectric device ofclaim 1, wherein the nanowire array comprises a plurality of individualnanowires.
 12. The thermoelectric device of claim 11, wherein theplurality of individual nanowires comprises one or more p-type elementsalternately connected to one or more n-type elements.
 13. Thethermoelectric device of claim 11, wherein the plurality of individualnanowires are connected electrically in series.
 14. The thermoelectricdevice of claim 11, wherein the plurality of individual nanowires areconnected in parallel.
 15. A thermoelectric device comprising at leastone p-doped nanowire array and at least one n-doped nanowire array,wherein the at least one p-doped nanowire array and the at least onen-doped nanowire array being connected electrically in series andthermally in parallel, wherein at least one nanowire array of the atleast one p-doped nanowire array and the at least one n-doped nanowirearray each comprises an individual nanowire consisting essentially ofdoped silicon, wherein the individual nanowire comprises asemiconductor, and wherein the individual nanowire has a diameter up toabout 20 nanometers (nm) and a length that is at least about 4 times thediameter of the individual nanowire, wherein the individual nanowire isdoped p-type or n-type, but not both, and has a doping concentrationfrom about 3×10¹⁹ cm′ to 2×10²⁰ cm⁻³, and wherein the individualnanowire has a thermoelectric efficiency that is at least one hundredtimes greater than a thermoelectric efficiency of bulk silicon, whereinthe thermoelectric efficiency of the individual nanowire is greater thanor equal to about 0.31 over a temperature range from about 100 K to 200K.
 16. The thermoelectric device of claim 15, wherein the at least onep-doped nanowire array comprises a plurality of p-doped semiconductornanowires and the at least one n-doped nanowire array comprises aplurality of n-doped semiconductor nanowires.
 17. The thermoelectricdevice of claim 16, wherein the p-doped semiconductor nanowires or then-doped nanowires have ZT efficiency values above 1, wherein${{ZT} = {\frac{S^{2}\sigma}{\kappa}T}},$ where S is the thermoelectricpower of the thermoelectric device, ${S = \frac{dV}{dT}},$ σ is theelectrical conductivity of the thermoelectric device, κ is the thermalconductivity of the thermoelectric device, and T is the temperature ofthe thermoelectric device.